Existence Results for a Fractional Boundary Value Problem via Critical Point Theory

Boucenna, A., and Moussaoui, Toufik. "Existence Results for a Fractional Boundary Value Problem via Critical Point Theory." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 54.1 (2015): 47-64. <http://eudml.org/doc/271600>.

@article{Boucenna2015,
abstract = {In this paper, we consider the following boundary value problem \[ \left\lbrace \begin\{array\}\{lll\} D\_\{T^\{-\}\}^\{\alpha \} (D\_\{0^\{+\}\}^\{\alpha \} (D\_\{T^\{-\}\}^\{\alpha \}(D\_\{0^\{+\}\}^\{\alpha \} u(t))) ) = f(t, u(t)), \quad t \in [0, T], \\ u(0)= u(T)= 0\\ D\_\{T^\{-\}\}^\{\alpha \}(D\_\{0^\{+\}\}^\{\alpha \}u(0))= D\_\{T^\{-\}\}^\{\alpha \}(D\_\{0^\{+\}\}^\{\alpha \}u(T))= 0, \end\{array\} \right. \] where $0 < \alpha \le 1$ and $f\colon [0, T]\times \mathbb \{R\} \rightarrow \mathbb \{R\} $ is a continuous function, $D_\{0^\{+\}\}^\{\alpha \}$, $D_\{T^\{-\}\}^\{\alpha \}$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.},
author = {Boucenna, A., Moussaoui, Toufik},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Existence results; fractional differential equation; boundary value problem; critical point theory; minimization principle; Mountain pass theorem; Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions},
language = {eng},
number = {1},
pages = {47-64},
publisher = {Palacký University Olomouc},
title = {Existence Results for a Fractional Boundary Value Problem via Critical Point Theory},
url = {http://eudml.org/doc/271600},
volume = {54},
year = {2015},
}

TY - JOUR
AU - Boucenna, A.
AU - Moussaoui, Toufik
TI - Existence Results for a Fractional Boundary Value Problem via Critical Point Theory
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2015
PB - Palacký University Olomouc
VL - 54
IS - 1
SP - 47
EP - 64
AB - In this paper, we consider the following boundary value problem \[ \left\lbrace \begin{array}{lll} D_{T^{-}}^{\alpha } (D_{0^{+}}^{\alpha } (D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha } u(t))) ) = f(t, u(t)), \quad t \in [0, T], \\ u(0)= u(T)= 0\\ D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(0))= D_{T^{-}}^{\alpha }(D_{0^{+}}^{\alpha }u(T))= 0, \end{array} \right. \] where $0 < \alpha \le 1$ and $f\colon [0, T]\times \mathbb {R} \rightarrow \mathbb {R} $ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.
LA - eng
KW - Existence results; fractional differential equation; boundary value problem; critical point theory; minimization principle; Mountain pass theorem; Third order; nonlinear differential equation; uniform stability; uniform ultimate boundedness; periodic solutions
UR - http://eudml.org/doc/271600
ER -